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PHYSICAL REVIEW D, VOLUME 59, 027101

The first space-based gravitational-wave detectors

R. R. Caldwell*Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 191

Marc Kamionkowski† and Leven Wadley‡

Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027~Received 28 July 1998; published 24 December 1998!

Gravitational waves provide a laboratory for general relativity and a window to energetic astrophysicalphenomena invisible with electromagnetic radiation. Several terrestrial detectors are currently under construc-tion, and a space-based interferometer is envisioned for launch early next century to detect test-mass motionsinduced by waves of relatively short wavelength. Very-long-wavelength gravitational waves can be detectedusing the plasma in the early Universe as test masses; the motion induced in the plasma by a wave is imprintedonto the cosmic microwave background~CMB!. While the signature of gravitational waves on the CMBtemperature fluctuations is not unique, thepolarizationpattern can be used to unambiguously detect gravita-tional radiation. Thus, forthcoming CMB polarization experiments, such as the Microwave Anisotropy Probeand Planck, will be the first space-based gravitational-wave detectors.@S0556-2821~99!01002-4#

PACS number~s!: 95.55.Ym, 04.80.Nn, 98.70.Vc

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One of the most spectacular predictions of general rtivity is the existence of gravitational waves. A gravitationwave conveys information about the motions of massripples in curvature—the shape of spacetime. Detectiongravitational radiation would allow us to probe ‘‘invisibleastrophysical phenomena hidden from view by absorptionelectromagnetic radiation. Observations of the binary puPSR1913116, which confirm the orbital-inspiral rate duethe emission of gravitational waves@1#, bring us tantalizinglyclose to this goal. However, we would still like to detegravitational radiation directly. Thus, a variety of efforts anow under way to detect gravitational waves@2#. Here, weshow that forthcoming maps of the polarization of the cmic microwave background~CMB! can be used to detecvery-long-wavelength gravitational radiation.

Gravitational waves are detected by observing the mothey induce in test masses@3,4#. High-frequency(1 – 104 Hz) gravitational waves, produced by the inspirand catastrophic collision of astrophysical objects, maydetectable by terrestrial laser interferometers@e.g., the LaserInterferometric Gravitational Wave Observatory~LIGO! @5##or resonant-mass antennae currently under construcLow-frequency (102421021 Hz) gravitational waves, produced by the orbital motion of binaries, could be detectedthe Laser Interferometer Space Antenna~LISA! @6#, a space-based interferometer targeted for launch circa 2015.

How can one detect ultra-low-frequency (10215

– 10218 Hz) gravitational radiation, with wavelengths comparable to the size of the observable Universe? The phobaryon fluid in the early Universe acts as a set of test mafor such waves. A gravitational wave in this frequency ranalternately squeezes and stretches the primordial plasmaas a resonant-mass detector is equipped with electronic

*Electronic address: caldwell@dept.physics.upenn.edu†Electronic address: kamion@phys.columbia.edu‡Electronic address: leven@phys.columbia.edu

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monitor the oscillation modes of the test body, the CMphotons are the electromagnetic signal upon which thplasma motions are imprinted.

One might despair because cosmological density perbations generate equivalent fluctuations in the CMB teperature, nullifying the possibility of using the temperatufluctuations to detect gravitational waves~although it can beused to place upper limits@7#!. To illustrate, we show twosimulated CMB temperature and polarization maps in FigThe color contrasts represent temperature fluctuationsroughly one part in 105 ~red are hot spots and blue are cold!.In Fig. 1~a!, the temperature fluctuations are produced bspectrum of stochastic cosmological density perturbatithat may be expected in a realistic inflationary cosmologiscenario. In Fig. 1~b! we have found a plausible spectrumstochastic long-wavelength gravitational waves that prodprecisely the same temperature fluctuations.

Yet our hopes are restored upon realization that the mtions in the cosmological fluid generated by gravitationwaves polarize the CMB in a pattern that is distinct from thproduced by density perturbations@8,9#. Roughly speaking,the polarization ‘‘vector’’ fieldPW (n) ~as a function of posi-tion n on the sky! can be decomposed into a curl and cufree part,

PW ~ n!5¹W A1¹W 3BW , ~1!

whereA is a scalar function andBW is a vector field. The curland curl-free parts ofPW (n) can be isolated by taking the cuand gradient ofPW , respectively. Since density perturbatioare scalar perturbations to the spacetime metric, they havhandedness and therefore produce no curl in the CMB poization field. Gravitational waves, however,do have a hand-edness, so theydo produce a curl. By decomposing the CMpolarization field into its curl and curl-free parts, one cunambiguously detect gravitational waves. Again, to illutrate, the headless arrows in Fig. 1 depict the orientation

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BRIEF REPORTS PHYSICAL REVIEW D 59 027101

magnitude of the polarization at each point on the sky.see that density perturbations and gravitational wavesproduce identical temperature maps can be distinguishethe polarization pattern.

More precisely, the Stokes parametersQ(n) andU(n), asa function of directionn5(u,f) on the sky, are componentof a symmetric trace-free~STF! 232 tensor,

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in spherical polar coordinates, with metricgab5diag(1,sin2 u). The polarization tensor can be expanded

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C are expansion coefficients@10#.

Consider a single gravitational wave with amplitudeh inthe early Universe,1 polarization, and comoving wave vec

FIG. 1. Simulated temperature-polarization maps of the CMsky for ~a! density perturbations and~b! long-wavelength gravita-tional waves. Notice that the temperature anisotropy maps, icated by the color patterns, are identical, e.g., the same cold,patches occur in both~a! and ~b!. However, the polarization patterns, indicated by the headless arrows, are distinct. For examthe orientation of the polarization vector along the border ofprojection maps differs in~a! and~b!. ~A color version of this figureis available at http://xxx.lanl.gov/abs/astro-ph/9807319.!

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tor k oriented in thez direction. The curl component of thpolarization pattern induced by this wave will have expasion coefficients

a~ l m!C,1 ~k,h!52~2p2i !hA2l 11~dm,22dm,22!

3DCl~T!~k,h51!. ~4!

The functionsDCl(T)(k,h51) describe the perturbations to a

isotropic photon distribution induced by a gravitational waof initially unit amplitude (h51); they are obtained fromequations for the photon distribution in an expanding Uverse with this gravitational wave. The precise formDCl

(T)(k,h51) is only weakly dependent on the cosmologicmodel. Onlym562 modes contribute since the effect of thgravitational wave is symmetric under a 180° rotation abthe direction of propagation.

Figure 2 shows the curl component of the polarization~aswell as the temperature fluctuation! produced by such agravitational wave, a record of the effect of the wave onspherical surface of last scatter. The direction of propagaas well as the polarization are clearly visible in this pattethe wavelength and the phase can also be inferred. Inregard, the CMB resembles a spherical resonant-mass dtor @11# more than it resembles LISA or LIGO.

What is the smallest dimensionless amplitudeh of agravitational wave of frequencyf that can be detected withsuch an experiment? Suppose a CMB polarization expment measuresQi and Ui at each ofi 51,2,. . . ,N smallregions on the sky, each of area 4p/N. Then we have 2Nindependent measurements ofh, one for eachQi and Ui ,each with an instrumental noises. The minimum-varianceestimator ofh for the map is obtained from the weighteaverage of all of these measurements, and the variancewhich h can be determined is thereforesh , given by

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In Fig. 3, we plot the results for the smallest amplitudes tcan be detected at the 2s level by the Microwave AnisotropyProbe~MAP! @12#, and the Planck Surveyor@13#, CMB sat-ellite experiments scheduled for launch, respectively,NASA in the year 2000 and by the European Space Agefive years later. The amplitude detectable by a given CMexperiment is proportional to the instrumental noises in thedetector. If we extrapolate the rate of progress in detectechnology the past few decades—roughly an order of mnitude per decade—into the future, then it is plausible tthe sensitivity could be improved by a factor of 100 over thof Planck within the timescale for flight of LISA. Thus, walso show in Fig. 3 the smallest amplitudeh that could bedetected by such a future CMB polarization experiment. Tsmallesth detectable by LIGO and LISA are also plotteFor reference, we show the amplitude of the largest scinvariant stochastic gravitational-wave background con

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BRIEF REPORTS PHYSICAL REVIEW D 59 027101

tent with the Cosmic Background Explorer~COBE! @7# ~theshort-dashed curve! and an upper limit from pulsar timing@14# ~the triangle!.

Are there any promising sources of such long-wavelengravitational radiation@15#? And if so, what could we learnfrom them? The most encouraging and intriguing sourcethe spectrum of gravitational waves produced from quanfluctuations in the spacetime metric~analogous to Hawkingradiation! during slow-roll inflation. If detected, these wavewould allow us to probe the early Universe well beyond tepoch when it became opaque to electromagnetic radiatioz.1100, out to the inflationary epoch at redshiftsz;1028,roughly 10220 seconds after the big bang. It can alsoshown that the amplitude of the stochastic gravitational-wbackground depends on the energy scale of inflation. Ifflation has something to do with grand unification, as matheorists surmise, then the amplitude should beh;1025,possibly within reach of these CMB experiments. Other psible sources of ultra-low-frequency gravitational wavesclude bubble collisions during a first-order phase transitin the early Universe@16# or the action of topological defect@17#. Thermal gravitational waves might be left over frothe Planck era@18#, and string theory-inspired alternativeand/or extensions to inflation predict a unique spectrumprimordial gravitational radiation@19,20#.

The primary stated goals of MAP and Planck will bedetermine the geometry of the Universe and the origin

FIG. 2. The temperature and curl component of the CMB poization pattern produced by a single gravitational wave with wanumberk54H0 /c, whereH0 is the Hubble constant, propagatinin the z direction with1 polarization. Density perturbations geneate no curl polarization. Hence, such a pattern is a unique signgravitational waves.~A color version of this figure is available ahttp://xxx.lanl.gov/abs/astro-ph/9807319.!

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large-scale structure. But as we have shown here, theseellite experiments, which will precede LISA by 10 to 2years, may also provide the first direct detection of gravtional radiation. The primordial plasma will provide the temasses for this detector, and these gravitational-wainduced motions are isolated unambiguously in the CMpolarization. Just as the early Universe is the poor maparticle accelerator, the CMB polarization will be the poman’s gravitational-wave detector. If these experiments rister a positive result, the implications for general relativiearly-universe cosmology, and quantum theory in curvspacetime will be staggering.

This work was supported at Columbia University by tU.S. DOE contract DEFG02-92-ER 40699, NASA ATgrant NAG5-3091, and the Alfred P. Sloan Foundation, aat the University of Pennsylvania by U.S. DOE contraDEFG02-95-ER 40893.

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FIG. 3. The smallest amplitudeh of a gravitational wave thatcan be detected at a frequencyf for LIGO, LISA, MAP, Planck,and a putative future CMB polarization experiment with 100 timthe Planck sensitivity. The short-dashed line shows the largscale-invariant stochastic gravitational-wave background consiswith COBE, and the triangle shows an upper limit from pulstiming.

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@1# J. H. Taylor, Rev. Mod. Phys.66, 711 ~1994!.@2# E.g., K. Thorne, inParticle and Nuclear Astrophysics an

Cosmology in the Next Millennium, Proceedings of the 1994Snowmass Summer Study, edited E. W. Kolb and R. D. Pe~World Scientific, Singapore, 1995!.

@3# H. Bondi, Nature~London! 179, 1072~1957!.@4# C. W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation

~Freeman, San Francisco, 1973!, pp. 444–455.@5# A. Abramovici et al., Science256, 325 ~1992!.

ei

@6# K. Danzmannet al., ‘‘Laser Interferometer Space AntennaPre-Phase A Report,’’ 1996.

@7# R. L. Davis et al., Phys. Rev. Lett.69, 1856 ~1992!; L. M.Krauss and M. White,ibid. 69, 869 ~1992!.

@8# M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys. ReLett. 78, 2058~1997!.

@9# U. Seljak and M. Zaldarriaga, Phys. Rev. Lett.78, 2054~1997!.

@10# M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys. Re

1-3

tro

-

ev.

ro,

BRIEF REPORTS PHYSICAL REVIEW D 59 027101

D 55, 7368~1997!.@11# W. W. Johnson and S. M. Merkowitz, Phys. Rev. Lett.70,

2367 ~1993!.@12# http://map.gsfc.nasa.gov.@13# http://astro.estec.esa.nl/SA-general/

Projects/Planck/@14# V. M. Kaspi, J. H. Taylor, and M. F. Ryba, Astrophys. J.428,

713 ~1994!.@15# B. Allen, in Proceedings of the Les Houches School on As

physical Sources of Gravitational Radiation, edited by J.-A.

02710

-

Marck and J.-P. Lasota~Cambridge University Press, Cambridge, England, 1997!.

@16# E.g., C. Baccigalupiet al., Phys. Rev. D56, 4610~1997!.@17# R. R. Caldwell, R. A. Battye, and E. P. S. Shellard, Phys. R

D 54, 7146 ~1996!; X. Martin and A. Vilenkin, Phys. Rev.Lett. 77, 2879~1996!.

@18# L. P. Grischuk, JETP40, 409 ~1974!.@19# G. Veneziano, hep-th/9802057.@20# Massimo Galluccio, Marco Litterio, and Franco Occhione

Phys. Rev. Lett.79, 970 ~1997!.

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